Liquid dispenser systems, such as liquid soap and sanitizer dispensers, provide a user with a predetermined amount of liquid upon actuation of the dispenser. In addition, it is sometimes desirable to dispense the liquid in the form of foam by, for example, injecting air into the liquid to create a foamy mixture of liquid and air bubbles. Many dispensers are refillable with refill units that comprise a pump (or a pump and an air compressor) and a container. Many of the refill units currently on the market are inverted. Many of the inverted refill units that have pumps dispense when the piston moves upward.